Materials, Science of

Title: Solving phase field equations using a meshless method

Authors: J X Zhou and M E Li

Source: Communications in numerical methods in engineering, Vol. 22, Issue 11, pp. 1109-1115.

Abstract:

The phase field equation is solved by using a meshless reproducing kernel particle method (RKPM) for the very first time. The 1D phase field equation is solved using different grid sizes and various time steps at a given grid size. The method can give accurate solutions across the interface, and allows a larger time step than explicit finite-difference method. The 2D phase field equation is computed by the present method and a classic shrinking of a circle is simulated. This shows the powerfulness and the potential of the method to treat more complicated problems.

Remember the C-source code for calculating the elastic stress fields of a circular cavity in a square plate under uniaxial stress? I have found another neat way of shipping code with comments (and equations and figures, if need be) thanks to Abi: here is the wikidot page of code, schematic and equations. Have fun!

Title: Intrinsic single-domain switching in ferroelectric materials on a nearly ideal surface

Authors: S. V. Kalinin, B. J. Rodriguez, S. Jesse, Y. H. Chu, T. Zhao, R. Ramesh, S. Choudhury, L. Q. Chen, E. A. Eliseev, and A. N. Morozovska

Source: PNAS December 18, 2007, vol. 104, no. 51, pp. 20204-20209

Abstract:
Ferroelectric domain nucleation and growth in multiferroic BiFeO3 is studied on a single-domain level by using piezoresponse force spectroscopy. Variation of local electromechanical response with dc tip bias is used to determine the size of the domain formed below the conductive scanning probe tip. The domain parameters are calculated self-consistently from the decoupled Green function theory by using tip geometry determined from the domain wall profile. The critical parameters of the nucleating domain and the activation energy for nucleation are determined. The switching mechanism is modeled by using the phase-field method, and comparison with experimental results shows that the nucleation biases are within a factor of $\latex \approx$ 2 of the intrinsic thermodynamic limit. The role of atomic-scale defects and long-range elastic fields on nucleation bias lowering is discussed. These measurements open a pathway for quantitative studies of the role of a single defect on kinetics and thermodynamics of first order bias-induced phase transitions and electrochemical reactions.

Title: Complexion: A new concept for kinetic engineering in materials science

Authors: Shen J. Dillon, Ming Tang, W. Craig Carter and Martin P. Harmer

Source: Acta Materialia, Vol. 55, Iss. 18, pp. 6208-6218 (2007)

Abstract:
Interfaces and the movement of atoms within an interface play a crucial role in determining the processing and properties of virtually all materials. However, the nature of interfaces in solids is highly complex and it has been an ongoing challenge to link material performance with the internal interface structure and related atomic transport mechanisms. Interface complexions offer a missing link to help solve this universal problem. We have theoretically predicted the existence of multiple interface complexions by thermodynamics, but the present work represents the most comprehensive characterization and proof of their existence in a real material system. An interface complexion can be considered as a separate phase, which can be made to transform into different complexions (phases) with vastly different properties by chemistry and heat treatment, thereby enabling the engineering control of material properties on a level not previously realizable. As such, complexions offer a solution to outstanding fundamental scientific mysteries, such as the origin of abnormal grain growth in inorganic materials, a problem which leading researchers in the field have struggled to explain for the past 50 years. It is also described how interface complexions will likely have widespread impact across all branches of material science and related disciplines.

Note: I have to read the paper carefully; at first glance, among other things, one of the claims seems to be an extension of Gibbsian definition of “phase” to account for interfaces too.

There are two important classes of models for the study of grain growth, namely, multiple order parameter models in which each allowed orientation is assigned an order parameter, and vector valued phase field models in which only a few order parameters are introduced. Here are the links to some papers that discuss vector valued phase field models:

1. Vector-valued phase field model for crystallization and grain boundary formation. R Kobayashi, J A Warren, and W C Carter, Physica D, 119, 415-423 (199
   

   We propose a new model for calculation of the crystalliztation and impingement of many particles with differing orientations. Based on earlier phase field models, a vector order parameter is introduced, and thus orientation of crystal/disordered interfaces can be determined relative to a crystalline frame. This model improves upon previous attempts to describe this phenomenon, as it requires far fewer equations of motion, and is energetically invariant under rotations. In this report a one-dimensional simulation of the model will be presented along with preliminary investigations of two-dimensional simulations.

2. A phase field model of the impingement of solidifying particles. J A Warren, W C Carter and R Kobayashi, Physica A, 261, 159-166 (199
   

   We propose a model of the impingement of solidifying crystalline particles, the ensuing grain boundary formation, and grain coarsening. This model improves upon previous theoretical descriptions of this phenomenon, in that it has the proper behavior under rotations and is easy to implement numerically. Also, insight into the model is straightforward since the parameters are physically motivated, and anisotropy in both the liquid–solid and grain boundary energies can be introduced in a natural manner. A one dimensional analytic solution is presented.

3. Modeling grain boundaries using a phase-field technique. J A Warren, R Kobayashi, and W C Carter, Journal of Crystal Growth, 211, 18-20 (2000)
   

   We propose a two-dimensional phase-field model of grain boundary dynamics. One-dimensional analytical solutions for a stable grain boundary in a bicrystal are obtained, and equilibrium energies are computed. By comparison with microscopic models of dislocation walls, insights into the physical accuracy of this model can be obtained. Indeed, for a particular choice of functional dependencies in the model, the grain boundary energy takes the same analytic form as the microscopic (dislocation) model of Read and Shockley (Phys. Rev. 78 (1950) 275).

4. A continuum model of grain boundaries. R Kobayashi, J A Warren, and W C Carter, Physica D, 140, 141-150 (2000)
   

   A two-dimensional frame-invariant phase field model of grain boundaries is developed. One-dimensional analytical solutions for a stable grain boundary in a bicrystal are obtained, and equilibrium energies are computed. With an appropriate choice of functional dependencies, the grain boundary energy takes the same analytic form as the microscopic (dislocation) model of Read and Shockley [W.T. Read, W. Shockley, Phys. Rev. 78 (1950) 275]. In addition, dynamic (one-dimensional) solutions are presented, showing rotation of a small grain between two pinned grains and the shrinkage and rotation of a circular grains embedded in a larger crystal.

5. Phase field model of premelting of grain boundaries. A E Lobkovsky, and J A Warren, Physica D, 164, 202-212 (2002).
   

   We present a phase field model of solidification which includes the effects of the crystalline orientation in the solid phase. This model describes grain boundaries as well as solid–liquid boundaries within a unified framework. With an appropriate choice of coupling of the phase field variable to the gradient of the crystalline orientation variable in the free energy, we find that high-angle boundaries undergo a premelting transition. As the melting temperature is approached from below, low-angle grain boundaries remain narrow. The width of the liquid layer at high-angle grain boundaries diverges logarithmically. In addition, for some choices of model coupling, there may be a discontinuous jump in the width of the fluid layer as function of temperature.

6. Nucleation and bulk crystallization in binary phase field theory. L Granasy, T Boerzsoenyi, and Pusztai, Physical Review Letters, 88, 20, 206105-1–206105-4 (2002)
   

   We present a phase field theory for binary crystal nucleation. In the one-component limit, quantitative agreement is achieved with computer simulations (Lennard-Jones system) and experiments (ice-water system) using model parameters evaluated from the free energy and thickness of the interface. The critical undercoolings predicted for Cu-Ni alloys accord with the measurements, and indicate homogeneous nucleation. The Kolmogorov exponents deduced for dendritic solidification and for “soft impingement” of particles via diffusion fields are consistent with experiment.

7. Extending phase field models of solidification to polycrystalline materials. J A Warren, R Kobayashi, A E Lobkovsky, and W C Carter, Acta Materialia, 6035-6058 (2003)
   

   We present a two-dimensional phase field model of grain boundary statics and dynamics. We begin with a brief description and physical motivation of the crystalline phase field model. The description is followed by characterization and analysis of several microstructural implications: the grain boundary energy as a function of misorientation, the liquid–grain–grain triple junction behavior, the wetting condition for a grain boundary and stabilized widths of intercalating phases at these boundaries, and evolution of a polycrystalline microstructure by solidification and impingement, followed by both grain boundary migration and grain rotation. Simulations that demonstrate these implications are presented, with a description of the numerical methods that were used to obtain them.

8. Equations with singular diffusivity. R Kobayashi and Y Giga, Journal of Statistical Physics, 95, 5/6, 1187-1220 (1999)
   

   Recently models of faceted crystal growth and of grain boundaries were proposed based on the gradient system with nondifferentiable energy. In this article, we study their most basic forms given by the equations and , where both of the related energies include a term of power one which is nondifferentiable at . The first equation is spatially homogeneous, while the second one is spatially inhomogeneous when depends on . These equations naturally express nonlocal interactions through their singular diffusivities (infinitely large diffusion constant), which make the profiles of the solutions completely flat. The mathematical basis for justifying and analyzing these equations is explained, and theoretical and numerical approaches show how the solutions of the equations evolve.

9. Sharp interface limit of a phase field model of crystal grains. A E Lobkovsky and J A Warren, Physical Review R, 63, 051605-1 — 051605-10 (2001)
   

   We analyze a two-dimensional phase field model designed to describe the dynamics of crystalline grains. The phenomenological free energy is a functional of two order parameters. The first one reflects the orientational order, while the second reflects the predominantally local orientation of the crystal. We consider the gradient flow of this free energy. Solutions can be interpreted as ensembles of grains (in which the orientation is constant in space) separated by grain boundaries. We study the dynamics of the boundaries as well as the rotation of the grains. In the limit of an infinitely sharp interface, the normal velocity of the boundary is proportional to both its curvature and its energy. We obtain explicit formulas for the interfacial energy and mobility, and study their behavior in the limit of a small misorientation. We calculate the rate of rotation of a grain in the sharp interface limit, and find that it depends sensitively on the choice of the model.

10. Phase field modeling of polycrystalline freezing. T Pusztai, G Bortel, and L Granasy, Materials Science and Engineering A, 413/414, 412-417 (2005)
    

    The formation of two and three-dimensional polycrystalline structures are addressed within the framework of the phase field theory. While in two dimensions a single orientation angle suffices to describe crystallographic orientation in the laboratory frame, in three dimensions, we use the four symmetric Euler parameters to define crystallographic orientation. Illustrative simulations are performed for various polycrystalline structures including simultaneous growth of randomly oriented dendritic particles, the formation of spherulites and crystal sheaves.

11. Phase field theory of polycrystalline solidification in three dimensions. T Pusztai, G Bortel, and L Granasy, Europhysics Letters, 71 (1), 131-137 (2005)
    

    A phase field theory of polycrystalline solidification is presented that describes the nucleation and growth of anisotropic particles with different crystallographic orientation in three dimensions. As opposed to the two-dimensional case, where a single orientation field suffices, in three dimensions, a minimum number of three fields are needed. The free energy of grain boundaries is assumed to be proportional to the angular difference between the adjacent crystals expressed here in terms of the differences of the four symmetric Euler parameters. The equations of motion for these fields are obtained from variational principles. Illustrative calculations are performed for polycrystalline solidification with dendritic, needle and spherulitic growth morphologies.

To give a short introduction to these papers:

Almost all the papers are related to solidification and the problem of impingement of different nuclei, which results in the grain structure when the solidification is complete. Thus, the problem of grain growth is incidental in all these papers; however, by modifying the bulk free energy density (by making sure that there is only one minimum which corresponds to the solid state), and dropping the thermal evolution equations (isothermal simulations), one can obtain equations that pertain to pure grain growth.

The idea behind papers 1-7, and 10-11 is that one can specify the crystalline orientations completely by giving an order parameter (say, ) which denoted the bulk of the grain (unity in the grain interior and less than unity at the grain boundaries), and an orientation parameter(s) field (say, , in the 2D case — Ref. 1-7, or, say, , in the 3D case, where represents an unit quaternion — Ref. 10-11). Ref.10-11 also show that the representation in terms of quaternion order parameters can be reduced to that of a single order parameter in 2D. These order parameters are evolved according to the Allen-Cahn equations meant for non-conserved order parameters.

While representing the orientation in terms of the quaternions or orientational order parameter , the bulk free energy of the system can only depend on the crystallanity parameter , and the gradients in and or , since the different orientations are all energetically favourable, and none is preferred over another.

In the following, for simplicity’s sake, let us consider a 2D model; the extension of the discussion to 3D is straightforward.

To obtain stable grain boundaries of finite width in thes models with orientational order parametes, we also need to introduce in the free energy, in addition to the usual terms.

The introduction of a term of the type leads to an evolution equation which contains a term of the type ; this leads to a singular diffusivity in the bulk of the grains since in the bulk is zero. While such a singular diffusivity allows for grain rotations in a natural manner in these models, it leads to both numerical and analytical difficulties.

The mathematical basis of dealing with singular diffusivities are dealt with in Ref. 8, while, the asymptotic analysis on these systems is performed in Ref. 9. And, Ref. 7, which is a review contains the details of the nuanced numerical implementations.

Finally, a couple of points that are problematic about these models (as far as my understanding of them goes):

(1) Ref. 1-7 and 9, deal with the problem as if the coordinate frame of reference used in the calculations is circular polar, which leads to extra terms of the type in the evolution equations. I believe they are extraneous, and should be dropped.

(2) The details of the addition or subtraction of terms in the Ref. 7 are again an artifact, I believe. In a true 3D case with quaternions (or a reduction thereof to 2D), such terms should not appear int he evolution equations.

(3) Though these models are capable of incorporating rotations, they might also lead to unphysical rotation events.

Before I end this post: soon, I will do a post on the other type of grain growth models with multiple order parameters, and how they compare with these vector order parameter models. I will also publish C codes of numerical implementation of these models. See you around!

The translation of a Dutch paper (PhD thesis?) of van der Waals into English by J S Rowlinson is available here (Unfortunately, I have no access to the soft copy version):

Van der Waals justifies the choice of minimization of the (Helmholtz) free energy as the criterion of equilibrium in a liquid-gas system (Sections 1–4). If density is a function of height h then the local free energy density differs from that of a homogeneous fluid by a term proportional to (d ² /dh ²); the extra term arises from the energy not from the entropy (Section 5). He uses this result to show how varies with h (Section 6), how this variation leads to a stable minimum free energy (Section 7), and to calculate the capillary energy or surface tension (Section 9). Near the critical point varies as ( _k -)^3/2, where _k is the critical temperature (Section 11). The paper closes with short discussions of the thickness of the surface layer (Section 12), of the difficulty of assuming that varies discontinuously with height (Section 14), and of the possible effect of derivatives of higher order than (d ² /dh ²) on the free energy and surface tension (Section 15).

Generally, the origin of the concept (and name) of spinodal decomposition is credited to van der Waals. I do not know if this document discusses that aspect of van der Waals’ work — I have to go to the library tomorrow.

Mathew Peet at Bainite has some pointers.

Title: Density functional studies of multiferroic magnetoelectrics

Author: Nicola A Hill

Source: Annual Review of Materials Research, Vol. 32: 1-37 (Volume publication date August 2002)

Abstract:

Multiferroic magnetoelectrics are materials that are both ferromagnetic and ferroelectric in the same phase. As a result, they have a spontaneous magnetization that can be switched by an applied magnetic field and a spontaneous polarization that can be switched by an applied electric field. In this paper we show that density functional theory has been invaluable both in explaining the properties of known magnetically ordered ferroelectric materials, and in predicting the occurrence of new ones. Density functional calculations have shown that, in general, the transition metal d electrons essential for magnetism reduce the tendency for off-center ferroelectric distortion. Consequently, an additional electronic or structural driving force must be present for ferromagnetism and ferroelectricity to occur simultaneously.

Everybody needs an iterative solver at some point or other. Recently, I sent a few links to a friend of mine. I thought I will also maintain the list in this page for future use.

• Templates for the solution of Linear systems: Building blocks for iterative methods. Richard Barrett, Michael Berry, Tony F. Chan, James Demmel, June Donato, Jack Dongarra, Victor Eijkhout, Roldan Pozo, Charles Romine, and Henk van der Vorst. SIAM
• Iterative methods for sparse linear systems. Yousef Saad. SIAM. (via the Wiki page on iterative methods).
• The iterative template library (with non-commerical use license);
• John Burkardt’s page is a very good place too, among many other things, for some of the template source codes (check out the FORTRAN codes);
• Netlib templates page with Matlab, C++, and Fortran codes; and,
• Templates for the solution of algebraic eigenvalue problems: a practical guide. Edited by Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe, and Henk van der Vorst. SIAM.

In case you know of any good resources that I have missed here, leave a note!

Title: Stability analysis of the Crank-Nicholson method for variable coefficient diffusion equation

Author: Charles Tadjeran

Source: Communications in Numerical Methods in Engineering, Vol 23, Issue 1, pp. 29-34, 2006.

Abstract:
The Crank-Nicholson method is a widely used method to obtain numerical approximations to the diffusion equation due to its accuracy and unconditional stability.

When the diffusion coefficient is not a constant, the general approach is to obtain a discretization for the PDE in the same manner as the case for constant coefficients. In this paper, we show that the manner of this discretization may impact the stability of the resulting method and could lead to instability of the numerical solution. It is shown that the classical Crank-Nicholson method will fail to be unconditionally stable if the diffusion coefficient is computed at the time gridpoints instead of at the midpoints of the temporal subinterval. A numerical example is presented and compared with the exact analytical solution to examine its divergence.